# Part - 2 Lecture - 5 Chapter 1 Relations and Functions

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5. Consider f : N → N, g: N → N and h: N → R defined as f (x) = 2x, g(y) = 3y + 4 and h(z) = sin z, ∀ x, y and z in N. Show that ho(gof) = (hog)of. (N)

6. Find gof and fog, if (N)
(i). $$f(x)=|x| \text{ and }g(x)=|5x-2|$$
(ii). $$f(x)=8x^3 \text{ and } g(x)=x^\frac{1}{3}$$

7. If f : R → R be given by$$f(x)=(3-x^3)^\frac{1}{3}$$, then find fof(x). (N)

8. Let$$g,f: R \rightarrow R$$ be defined by $$f(x)=2x+1$$ and $$g(x)=x^2-2,\forall x\in R$$, respectively. Then, find gof. (E)

9. Let $$g,f: R\rightarrow R$$ be defined by $$g(x)=\frac{x+2}{3},f(x)=3x-2$$ write fog(x). (B)

10. If f : R → R defined by $$f(x)=\frac{x-1}{2}$$, find (fof)(x). (B)

11. If f : R → R defined by $$f(x)=x^2-3x+2$$, find (fof)(x). (E)

12. If $$f(x)=log{\left(\frac{1+x}{1-x}\right)}$$, show that$$f\left(\frac{2x}{1+x^2}\right)=2f(x)$$. (B)

13. Let $$f: R \rightarrow R$$ be defined by $$f(x)=x^2+1$$ , find the pre image of 17 and -3. (B)

14. If $$f: R \rightarrow R, g: R \rightarrow R$$, given by $$f(x)=[x],g(x)=|x|$$, then find $$fog\left(-\frac{2}{3}\right) \text{ and } gof\left(-\frac{2}{3}\right)$$. (B)